| L(s) = 1 | − 1.43·3-s − 2.51·5-s + 7-s − 0.950·9-s − 1.95·11-s + 5.38·13-s + 3.60·15-s − 3.95·17-s − 19-s − 1.43·21-s − 8.51·23-s + 1.34·25-s + 5.65·27-s + 3.95·29-s + 5.95·31-s + 2.79·33-s − 2.51·35-s + 7.55·37-s − 7.70·39-s + 6.81·41-s + 5.03·43-s + 2.39·45-s + 5.65·47-s + 49-s + 5.65·51-s + 11.4·53-s + 4.91·55-s + ⋯ |
| L(s) = 1 | − 0.826·3-s − 1.12·5-s + 0.377·7-s − 0.316·9-s − 0.588·11-s + 1.49·13-s + 0.931·15-s − 0.958·17-s − 0.229·19-s − 0.312·21-s − 1.77·23-s + 0.268·25-s + 1.08·27-s + 0.733·29-s + 1.06·31-s + 0.486·33-s − 0.425·35-s + 1.24·37-s − 1.23·39-s + 1.06·41-s + 0.768·43-s + 0.356·45-s + 0.824·47-s + 0.142·49-s + 0.791·51-s + 1.57·53-s + 0.662·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7923687586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7923687586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.43T + 3T^{2} \) |
| 5 | \( 1 + 2.51T + 5T^{2} \) |
| 11 | \( 1 + 1.95T + 11T^{2} \) |
| 13 | \( 1 - 5.38T + 13T^{2} \) |
| 17 | \( 1 + 3.95T + 17T^{2} \) |
| 23 | \( 1 + 8.51T + 23T^{2} \) |
| 29 | \( 1 - 3.95T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 37 | \( 1 - 7.55T + 37T^{2} \) |
| 41 | \( 1 - 6.81T + 41T^{2} \) |
| 43 | \( 1 - 5.03T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 0.617T + 59T^{2} \) |
| 61 | \( 1 - 5.48T + 61T^{2} \) |
| 67 | \( 1 + 0.223T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 7.43T + 73T^{2} \) |
| 79 | \( 1 + 5.03T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 4.69T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.21844940853996245343386289422, −8.771638859501805644182855689416, −8.274027970813317211260359924162, −7.50304514719632252226127060544, −6.28998986792017607383278570623, −5.80599505797635897354797308399, −4.51959219564231288894304882417, −3.94705011126135679373494749854, −2.52545108024124213787741884927, −0.69880978290919969296941795755,
0.69880978290919969296941795755, 2.52545108024124213787741884927, 3.94705011126135679373494749854, 4.51959219564231288894304882417, 5.80599505797635897354797308399, 6.28998986792017607383278570623, 7.50304514719632252226127060544, 8.274027970813317211260359924162, 8.771638859501805644182855689416, 10.21844940853996245343386289422
